The Sherrington-Kirkpatrick spin glass model in the presence of a random field with a joint Gaussian probability density function for the exchange interactions and random fields

0.35 MB·

by  Ioannis A. Hadjiagapiou

The Sherrington-Kirkpatrick spin glass model
in the presence of a random field with a joint
Gaussian probability density function for the
4
1 exchange interactions and random fields
0
2

n
a
J
0 Ioannis A. Hadjiagapiou∗
1
Section of Solid State Physics, Department of Physics, University of Athens,

] Panepistimioupolis, GR 15784 Zografos, Athens, Greece
h
c
e
m
- Abstract
t
a
t
s The magnetic systems with disorder form an important class of systems, which
.
at are under intensive studies, since they reflect real systems. Such a class of systems
m is the spin glass one, which combines randomness and frustration. TheSherrington-
- KirkpatrickIsingspinglasswithrandomcouplingsinthepresenceofarandommag-
d
netic field is investigated in detail within the framework of the replica method. The
n
o two randomvariables (exchange integral interaction andrandommagnetic field)are
c
drawn from a joint Gaussian probability density function characterized by a corre-
[
lation coefficient ρ. The thermodynamic properties and phase diagrams are studied

1
with respect to the natural parameters of both random components of the system
v
6 contained in the probability density. The de Almeida-Thouless line is explored as
4 a function of temperature, ρ and other system parameters. The entropy for zero
2
temperature as well as for non zero temperatures is partly negative or positive,
2
. acquiring positive branches as h increases.
1 0
0
4
Key words: Ising model, spin glass, frustration, replica method, random field,
1
: Gaussian probability density
v
i PACS: 64.60.De, 05.70.Fh, 75.10.Nr, 75.50.Lk
X
r
a
Corresponding author.
∗
Email address: [email protected] (Ioannis A. Hadjiagapiou).
Preprint submitted to Physica A 13 January 2014
1 Introduction
The critical properties of magnetic systems with quenched disorder has been
a topic of growing interest in statistical physics over the last years and still
attracts considerable attention in spite of their long history, since it has been
established that the introduction of randomness can cause important effects
on their thermodynamic behavior in comparison to the pure ones; the type
of the phase transition as well as the universality class may change [1,2,3]. In
two-dimensions an infinitesimal amount of disorder converts a second-order
phase transition (SOPT) into a first-order phase transition (FOPT), whereas
in three-dimensions it is converted to an FOPT only when disorder exceeds
a threshold. Randomness is encountered in the form of vacancies, variable
or diluted bonds, impurities [4,5], random fields [6,7,8,9,10,11,12,13] and spin
glasses [13,14,15,16,17,18,19]. In addition, the study of disordered systems
is a necessity because homogeneous systems are, in general, an idealization,
whereas real materials contain impurities, nonmagnetic atoms or vacancies
randomly distributed within the system consisting of magnetic atoms, or vari-
able bonds in magnitude and/or in sign; such systems have attracted wide
interest and have been studied intensively theoretically, numerically and ex-
perimentally, since it is a matter of great urgency to develop an understanding
of the role of these non ideal effects. However, to facilitate their study, these
aremodeled by relying onpure-system models modifiedaccordingly, e.g. Ising,
Potts, Baxter-Wu, etc.
An important manifestation of disorder is the presence of random magnetic
fields acting on each spin of the magnetic system under consideration in an
otherwise free of defects lattice, whose pure version is modeled according to a
current one, e.g. the Ising model; the system now in the presence of such fields
is called random field Ising model (RFIM) [6,7,8,9,10,11,12,13]. Associated
with this model are the notions of lower critical dimension, tricritical points,
scaling laws, crossover phenomena, higher order critical points and random
field probability distribution function (PDF). RFIM had been the standard
vehicle for studying the effects of quenched randomness on phase diagrams
and critical properties of lattice spin systems and had been studied for many
years since the seminal work of Imry and Ma [11]. The RFIM Hamiltonian, in
the case of constant exchange integral, is
H = J S S h S , S = 1 (1)
i j i i i
− − ±
<i,j> i
X X
This Hamiltonian describes the competition between the long-rangeorder (ex-
pressed by the first summation) and the random ordering fields. We also con-
siderthatJ > 0sothatthegroundstateisferromagneticinabsenceofrandom
fields. The RFIM model, despite its simple definition and apparent simplicity,
2
together with the richness of the physical properties emerging from its study,
hasmotivateda significant number ofinvestigations; however, these properties
have proved to be a source of much controversy, primarily due to the lack of a
reliable theoretical foundation. Still, substantial efforts to elucidate the basic
problems of the RFIM continue to attract considerable attention, because of
its direct relevance to a number of significant physical problems. The presence
of random fields requires two averaging procedures, the usual thermal average,
denoted by angular brackets ... , and disorder average over the random fields
h i
denoted by ... for the respective PDF, which is usually a version of the
r
h i
bimodal, trimodal or Gaussian distributions. The most frequently used PDF
for random fields is either the bimodal or the single Gaussian; the former is,
P(h ) = pδ(h h )+qδ(h +h ) (2)
i i 0 i 0
−
where p is the fraction of lattice sites having a magnetic field h , while the
0
rest sites have a field ( h ) with site probability q such that p + q = 1; the
0
−
usual choice was p = q = 1, symmetric case. The latter PDF is,
2
1 h2
P(h ) = exp i (3)
i (2πσ2)1/2 "−2σ2#
with zero mean and standard deviation σ.
Oneof themain issues was theexperimental realization ofrandomfields. Fish-
man and Aharony [20] showed that the randomly quenched exchange interac-
tions Ising antiferromagnet in a uniform field H is equivalent to a ferromagnet
ina randomfield with thestrength of therandom field linearly proportionalto
the induced magnetization. This identification gave new impetus to the study
of the RFIM, the investigation gained further interest and was intensified re-
sulting in a large number of publications (theoretical, numerical, Monte Carlo
simulations and experimental) in the last thirty years. Although much effort
had been invested towards this direction, the only well-established conclusion
drawn was the existence of a phase transition for d 3 (d space dimension),
≥
that is, the critical lower dimension d is 2 after a long controversial discus-
l
sion [11,21], while many other issues are still unanswered; among them is the
order of the phase transition (first or second order), the universality class and
the dependence of these points on the form of the random field PDF. Galam,
via MFA, has shown that the Ising antiferromagnets in a uniform field with
either a general random site exchange or site dilution have the same multi-
critical space as the random-field Ising model with bimodal PDF [22]. The
study of RFIM has also highlighted another feature of the model, that of
tricriticality and its dependence on the assumed distribution function of the
random fields. A very controversial issue has arisen concerning the effect of
the random-field probability distribution function on the equilibrium phase
3
diagram of the RFIM. The choice of the random field distribution can lead to
a continuous ferromagnetic/paramagnetic (FM/PM) boundary as in the sin-
gle Gaussian PDF, whereas for the bimodal one this boundary is divided into
two parts, an SOPT branch for high temperatures and an FOPT branch for
low temperatures separated by a tricritical point (TCP) at kTt/(zJ) = 2/3
c
and ht/(zJ) = (kTt/(zJ)) argtanh(1/√3) 0.439 [12], where z is the co-
c c × ≃
ordination number and k the Boltzmann constant, such that for T < Tt and
c
h > ht the transition to the FM phase is of first order for p = 1. However,
c 2
this behavior is not fully elucidated since in the case of the three-dimensional
RFIM, the high temperature series expansions by Gofman et al [23] yielded
only continuous transitions for both probability distributions, whereas accord-
ing to Houghton et al [24] both distributions predicted the existence of a TCP,
with ht = 0.28 0.01andTt = 0.49 0.03forthe bimodal andσt = 0.36 0.01
c ± c ± c ±
and Tt = 0.36 0.04 for the single Gaussian. In the Monte Carlo studies for
c ±
d = 3, Machta et al [25], using single Gaussian distribution, could not reach a
definite conclusion concerning the nature of the transition, since for some real-
izationsofrandomnessthemagnetizationhistogramwastwo-peaked(implying
an SOPT) whereas for other ones three-peaked, implying an FOPT; Middle-
ton and Fisher [26], using a similar distribution for T = 0, suggested an SOPT
with a small order parameter exponent β = 0.017(5). Malakis and Fytas [27],
by applying the critical minimum-energy subspace scheme in conjunction with
the WangLandau and broad-histogram methods for cubic lattices, proved that
the specific heat and susceptibility are non-self-averaging using the bimodal
distribution.
Another notable manifestation of randomness is the spin glass (SG) phase
exhibited by many systems under certain conditions. These are random mag-
netic systems in which the interactions between the spins are in conflict to
each other, a phenomenon known as frustration, a result of strong frozen-in
structural disorder according to which no single spin configuration is favored
by all interactions, quenched randomness. Moreover, SG is an emergent phase
of matter in random magnetic systems and thus a lot of studies on disordered
systems concern that phase, where the magnetic and non-magnetic compo-
nents, making up the material, are randomly distributed in space; the disorder
is present in the Hamiltonian in the form of random couplings between two
constituent spins, which vary, in general, in their values and signs according
to a PDF P(J ) chosen suitably [13,14,15,16,17,18,19]. The competing inter-
ij
actions are ferromagnetic and antiferromagnetic. Conventional SGs are dilute
magnetic alloys such as AuFe or CuMn. The main objective is to understand
better, at a theoretical level, what are the microscopic mechanisms leading to
such a behavior and how to describe them. Many studies, mainly using the
mean-field analysis, have been successful in elucidating various concepts for
understanding SGs. One of the current issues in SGs is their nature in finite
dimensions below the upper critical dimension. Unfortunately, for finite di-
mensions, the calculations often rely on numerical simulations, because there
4
are few ways to analytically study SGs. Long equilibration times for their nu-
merical simulations are needed and average over many realizations of random
systems to make error bars small enough. It is thus difficult to gain a conclu-
sive understanding on the nature of them in finite dimensions. Establishing
reliable analytical theories of SGs have been one of the most challenging prob-
lems for years. Theoretical physicists have developed mathematically heuristic
tools based on what is called the “replica trick”. Another successful analysis
to elucidate their properties is the use of gauge symmetry, by which one can
obtain the exact value of the internal energy, evaluate the upper bound for the
specific heat, and obtain some correlation inequalities in a subspace known as
the Nishimori line [28]. Since the Nishimori line is also invariant under renor-
malization group transformations, the intersection of the Nishimori line and
the FM/PM transition line must be a fixed point. The so-called Nishimori
point corresponds to a new universality class belonging precisely to the fam-
ily of strong disorder fixed points. Moreover, Kaneyoshi has also applied the
effective field theory to the SGs [29].
In the parameter plane spanned by temperature and external magnetic field
the high temperature phase is separated from the spin glass one by the so
called de Almeida-Thouless line (AT-line); consequently, the determination of
the AT-line is a matter of great urgency in the theoretical analysis of the SGs
model. The equilibrium properties of mean field SGs are calculated by using
the two available different approaches. The first one is the replica method
starting with n replicas of the system under consideration. The free energy
canbedetermined bythesaddle pointmethod. Themainfeatureofthereplica
method is that the mathematically problematic limit n 0 is usually taken
→
at the end. In this framework, the AT-line is determined by the local stability
of the replica symmetric saddle point. In the other method, the cavity one,
one spin is added to a system of N spins and a stochastic stability of the
thermodynamic limit N is used to derive self-consistent equations for
→ ∞
the order parameters. In the latter procedure, the AT-line is obtained by
investigating the correlations between two spins which vanish in the thermal
limit for a pure state of a mean field system [30,31,32,33,34,35,36,37].
AnearlyattemptinthetheoryofSGswasputforwardbyEdwardsandAnder-
son (EA) [30], based on Ising model with the disorder in the exchange integral
between nearest neighbors; they managed to demonstrate the existence of the
spin glass phase within the mean field theory in conjunction with the replica
trick; they also identified two features for a spin glass theory, frustration and
disorder. As their main result was the introduction of a new type of ”order”
parameter, which describes the long-time correlations q =<< S2 > > = 0,
i T J6
where < > means configuration averaging over the distributions P(J ) for
J ij
all spin pairs (ij) and < > means thermal averaging. A simple mean field
T
approximation leads to q(T) = 0 below a characteristic temperature T and to
f
6
a sharp second order phase transition at T . Their model was a generalization
f
5
of the Ising model (Ising spin-glass, ISG) but with a non constant exchange
integral interaction, namely,
H = J S S , S = 1 (4)
ij i j i
− ±
<i,j>
X
where < i,j > implies summation over nearest neighbors, J is the bilin-
ij
ear exchange interaction between nearest-neighbor pairs, randomly quenched
variables, identically and independently distributed according to the single
Gaussian probability distribution function
P(J ) = [(2π)1/2J] 1exp J2/(2J2) (5)
ij − − ij
h i
with zero mean value and variance J2. The disorder is quenched, in that J ,
ij
initially, are chosen randomly but then fixed for all thermodynamic processes.
The ISG together with the RFIM constitute two of the most-studied subjects
in the area of the disordered magnetic systems. The EA model is far too dif-
ficult to be analyzed theoretically in detail, thus Sherrington and Kirkpatrick
(SK) [31] introduced in 1975 a simplified version of this model by replacing
the pair interaction by a long-range one, in that, each particle interacts with
the remaining ones, so that the Hamiltonian (4) is replaced by
1
H = J S S , S = 1 (6)
ij i j i
−2 ±
X(ij)
where (ij) implies summation over all pairs of spins; the bilinear spin inter-
actions J are randomly quenched variables, specified by a symmetric matrix
ij
J and are distributed according to the probability distribution function
ij
{ }
P(J ) = [(2π)1/2J] 1exp (J J )2/(2J2) (7)
ij − ij 0
− −
h i
with J and J scaled by,
0
J = J/N1/2,J = J /N (8)
0 0
e e
bothJ andJ areintensive quantities, so thatintheSKmodel each spininter-
0
acts with each other one via weak interaction of the order N 1/2. The presence
−
e e
of N (the total number of spins in the system) is necessary to ensure that the
respective thermodynamic quantities are extensive. This simple model cap-
tures the basic ingredients of spin glass physics, namely, quenched random-
ness and frustration, and was solved “exactly” at the mean field level. The
fluctuations around the SK saddle point are described by an (n(n−1) n(n−1))
2 ∗ 2
6
matrix and its eigenvalues have been determined inRef. [36]. The temperature
dependence of these eigenvalues shows that the replica symmetric saddle point
loses its stability at the phase boundary of the SG phase. The SK model, since
its inception, is still alive and attractive presenting several challenging issues;
it presents a continuous phase transition in which the spin glass phase has the
free energy landscape composed by many almost degenerated thermodynamic
states separated by infinitely high barriers. Its formalism has gone far beyond
the area of disordered magnetic systems being employed in many other com-
plex systems, like neural networks, optimization problems as well as in stock
markets and in wireless network communications [38]. Many experimental ob-
servationsseemtobeingoodagreement withthepredictionsofthismodel.An
important suggestion concerning the behavior of various physical parameters
in the SG phase is the so-called Parisi-Toulouse (PaT) hypothesis, or projec-
tion hypothesis, for the SK model, according to which the entropy and the
EA order parameter q are field independent S(T,h) = S(T),q(T,h) = q(T) ,
whereas magnetization is temperature indep(cid:16)endent m(T,h) = m(h) , wher(cid:17)e
h is an external magnetic field; Monte Carlo results s(cid:16)how strong evide(cid:17)nce that
entropy is independent of the applied field, [39,40,41,42,43].
In addition to the SK model other infinite-range spin glass models have been
proposed, including Blume-Emery-Capel, Potts, spin-S (S > 1/2) and vector
spin glass models [14,18,19]. New phases, characterized by different classes of
order parameters, have emerged, opening many controversial problems from
both theoretical and experimental points of view.
The paper is organized as follows. In the next section, we discuss the lim-
iting procedure between the canonical partition function and the Helmholtz
free energy, the replica approach and suggest the joint Gaussian PDF with
correlation function ρ. In section 3, we introduce the current model with the
random field and calculate the respective free energy functional as well as the
magnetization and the Edwards-Anderson parameter. In section 4 we present
the numerical results, the phase diagram and other thermodynamic quantities
and we close with the conclusions and discussions in section 5.
2 The free energy and replica approach
As in every problem in equilibrium statistical physics the central issue is the
calculation of the free energy per particle from the respective one of the N-
particle system F(β,N) in the thermodynamic limit, namely
1
f(β) = lim F(β,N) (9)
N N
→∞
where βF(β,N) = lnZ(β,N), Z(β,N) is the canonical partition function.
−
However, aftertheintroductionofrandomness,asinthecaseofspinglasses,its
7
influence on the system has to be considered, so that the relation (9) converts
into,
1
βf(β) = lim lnZ(β,N) (10)
− N N r
→∞ D E
where ... represents the thermal average as well as the one with respect to
h ir
the randomness, thus the calculation of the free energy per particle is trans-
ferred to the calculation of lnZ(β,N) .
h ir
The eventual aimis to calculate the variousobservables, but to achieve this we
calculate initially the Helmholtz free energy F; the calculation of the partition
function is a very hard task, resulting the need for a new procedure to make
the calculation feasible; as far as the function needed is the logarithm of the
partition function and not the partition function itself, the following formula
can be applied,
Zn 1
lnZ = lim − (11)
n 0 n
→
implying thatwehaveconsidered nreplicasoftheinitialsystem, which arenot
interacting, and Zn = n Z , where α is the replica identifier and instead
α=1 α
of averaging lnZ we average Zn. Considering this expression for lnZ , the
Q h i
relation (10) converts into
1
βf(β) = lim lim Zn 1 (12)
− n 0N Nn r −
→ →∞ (cid:16)D E (cid:17)
the order of the limits is irrelevant [32], although the thermodynamic one
precedes that of n 0 in order to apply the steepest descent method; the
→
replicated partition function for integer values of n assumes the form
n
Zn(β) = exp β H Sα (13)
− { i }
{SiαX=±1} h αX=1 (cid:16) (cid:17)i
Sherrington and Kirkpatrick managed to derive an expression for the respec-
tive free energy, by calculating initially the free energy functional with respect
to the two parameters m and q dependent on replicas, introduced through
α αγ
the Hubbard-Stratonovich transformation, with α and γ characterizing the
replicas and α = γ, among other parameters; by considering the replica sym-
6
metry hypothesis (RS), that is m = m and q = q for every α and γ, and
α αγ
using the analytic continuation n 0 succeeded in calculating the system
→
8
free energy, namely
J2β2 J m2β 1
βF = N (1 q)2 + 0 dze z2/2ln 2cosh(H(z)) (14)
−
(− 4 − 2 − (2π)1/2 )
e e Z h i
f
where H(z) = βJq1/2z + βJ m, m =<< S > > and q =<< S >2> , in
0 i T J i T J
conjunction with the exponential identity (Hubbard-Stratonovich transforma-
f e e
tion)
1/2
2 λ ∞ 2
eλσ2 = e−λx2 +λσxdx (15)
2π!
Z
−∞
The SK model was solved by means of the replica method and, consequently,
it was originally thought that it was exactly solvable, although Sherrington
and Kirkpatrick were aware that it suffers from a serious drawback, in that, it
possesses a negative entropy at zero temperature, specifically S(T = 0 K) =
◦
Nk/2π. SK model was the subject of a large number of publications and
−
cannot be considered as a trivial one in the mean field sense. Though rather
unrealistic, it seems to describe some spin glass properties correctly. The SK
model for Ising spins with quenched random bonds is the simplest representa-
tive of a class of long-ranged models all successfully describing the interesting
phenomena of spin glasses. In addition to this success in physical questions,
the research on these models has been fruitful and stimulating in optimiza-
tion problems, in understanding the neural networks and communications.
The zero-temperature entropy anomaly (s(T = 0) < 0), appearing in the SK
model, has been shown to be associated with the hypothesis of the replica
symmetry of the two order parameters, m and q. The correct low-temperature
solution, resulting by breaking the replica symmetry of SK, was proposed by
Parisi [34], and consists of a continuous order parameter function (an infinite
number of order parameters) associated with many low-energy states, a pro-
cedure which is usually called the replica-symmetry breaking (RSB). Parisi
was the first who found a satisfactory solution of the SK model in the SG
regime. Curiously, the simplest one-step RSB (1S-RSB) procedure improves,
in part, this anomaly, in the sense that the zero temperature entropy per
particle becomes less negative, from s(0)/(Nk ) 0.16 within the RS it
B
≃ −
rises to s(0)/(Nk ) 0.01 within the 1S-RSB, a significant improvement.
B
≃ −
The complementary approach by Thouless, Anderson and Palmer (TAP), for
investigating the spin glass model, does not perform the bond average, and
permits a treatment of problems depending on specific configurations [35].
For other questions which are expected to be independent of the special con-
figuration, such as all macroscopic physical quantities, self averaging occurs.
This is due to the fact that the random interaction matrices have well-known
asymptotic properties in the thermodynamic limit. The situation is in prin-
9
ciple similar to the central limit theorem in probability theory, where large
numbers of random variables also permit the calculation of macroscopic quan-
tities which hold for nearly every realization of the random variables. Thus
the investigation of one or some representative systems is sufficient and the
bond average is not needed. The TAP equations have been well established
for more than two decades and several alternative derivations are known. Nev-
ertheless the TAP approach is still a field of current interest. This is due to
the importance of the approach to numerous interesting problems. Moreover
it is suspected that not all aspects of this approach have yet been worked
out. Furthermore, a transition in the presence of an external magnetic field,
known as the Almeida-Thouless (AT) line [36,37], is found in the solution of
the SK model: such a line separates a low-temperature region, characterized
by RSB, froma high-temperature one, where a simple one-parameter solution,
RS solution, is stable. Numerical simulations are very hard to carry out for
short-range ISGs on a cubic lattice, due to large thermalization times; as a
consequence, no conclusive results in three-dimensional systems are available.
However, in four dimensions the critical temperature is much higher, making
thermalization easier; in this case, many works claim to have observed some
mean-field features.
Reentrant spin-glass (RSG) transition is a well-known phenomenon of spin
glasses. The RSG transition is found near the phase boundary between the
SG phase and the FM phase. As the temperature decreases from a higher
temperature, magnetization once increases and then disappears at a lower
temperature. Finally, the SG phase is realized. The phenomenon was first con-
sidered as a phase transition between an FM phase and a SG phase. However,
neutron diffraction studies have revealed that the SG phase is characterized
by FM clusters. Now the RSG transition is believed to be a reentry from a
FM phase to a frozen state with FM clusters. The mechanism responsible
for this reentrant transition has not yet been resolved. Two ideas have been
proposed for describing the RSG transition: (i) an infinite-range Ising bond
model, and (ii) a phenomenological random field concept. The essential point
of that conception is that the system is decomposed into an FM part and
a part with frustrated spins (SG part). At low temperatures, the spins of
the SG part yield random effective fields to the spins of the FM part. Nev-
ertheless, no theoretical evidence has yet been presented for this idea in a
microscopic point of view. In the last two decades, computer simulations have
been performed extensively to solve theRSG transition in various models such
as short-range bond models [18,10,11], short-range site models [12,20], and a
Ruderman-Kittel-Kasuya-Yoshida model [21,22].
Althoughtheaforementionedtypes ofrandomness inmagneticsystems consist
a significant branch of statistical physics, very few investigations have consid-
ered them together [44], but even then the considered probability density
function for the random bonds and random fields were considered as distinct
10